2021 ᠣᠨ ᠤ ᠤᠯᠠᠭᠠᠨᠬᠠᠳᠠ ᠬᠤᠲᠠ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠳᠤ ᠰᠢᠯᠭᠠᠯᠲᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ

20 21 ᠣᠨ ᠤ ᠤᠯᠠᠭ ᠠᠨᠬᠠᠳᠠ ᠬᠣᠲ ᠠ ᠶᠢᠨ ᠠᠩᠬ ᠠ ᠳᠤᠮ ᠳᠠᠳᠤ ᠶᠢ ᠨ ᠰ ᠤᠷᠤᠭ ᠴᠢᠳ ᠤᠨ ᠰᠤᠷᠤᠯ ᠭ ᠠ ᠲᠡᠭᠦᠰ ᠬᠦ ᠂ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠳᠠ ᠪᠰᠢᠬ ᠤ ᠨᠢᠭᠡᠳ ᠤᠯᠲᠠᠢ ᠰᠢ ᠯᠭᠠᠯᠲ ᠠ ᠶᠢᠨ ᠰᠡᠲᠤᠪ ᠲᠣᠭᠠᠨ ᠤ ᠤᠬᠠᠭᠠᠨ ᠨᠢᠭᠡ᠂ ᠰᠤᠩᠭᠤ ᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ( ᠰ ᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬ ᠤᠪᠢ ᠂ ᠨᠠ ᠢᠲᠡ 42 ᠬᠤ ᠪᠢ) 1. −2021 ᠶᠢᠨ ᠡᠰᠡᠷ ᠬᠦ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 2021 B. −20 21 C. 12021 D. − 12021 2. ᠪᠡᠭᠡᠵᠢᠩ ᠦᠨ ᠴᠠᠭ ᠢᠶᠠᠷ 20 21 ᠣ ᠨ ᠤ 1 ᠳ᠋ᠦᠭᠡᠷ ᠰᠠᠷ ᠠ ᠶᠢᠨ 3 ᠦ ᠡᠳᠤᠷ ᠤᠨ 6 ᠴᠠ ᠭ ᠪᠣᠯᠲᠠ ᠯ ᠠ᠂ ᠮᠠᠨ ᠤ ᠤᠯᠤᠰ ᠤ ᠨ ᠠ ᠩᠬᠠᠳᠠ ᠭᠴᠢ ᠤᠳᠠ ᠭᠠᠨ ᠤ ᠭᠠᠯ ᠭᠠᠷᠠᠭ ᠢ ᠬᠠᠢᠨ ᠪᠠᠢᠴᠠᠭ ᠠᠬᠤ ᠡᠭᠦᠷ ᠭᠡ ᠶᠢ ᠡᠭᠦᠷ ᠭᠡᠯᠡᠵᠦ ᠪᠠᠢᠭ ᠠ ︽ᠲᠢᠶᠡᠨ ᠸᠧ ᠨ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠨᠣᠮ ᠧᠷᠲᠤ ︾ ᠭᠠᠯ ᠭᠠᠷᠠᠭ ᠤᠨ ᠬᠠᠢᠨ ᠪᠠᠢᠴᠠᠭ ᠠᠬᠤ ᠲᠥᠬᠥ ᠭᠡᠷᠥ ᠮᠵᠢ ᠨᠢᠭᠡᠨᠲᠡ ᠵᠢᠮ ᠢᠶᠡᠷ ᠢᠶᠡᠨ ᠪᠠᠷᠤᠭ 163 ᠡᠳᠤᠷ ᠠᠶ ᠠᠯᠠᠵᠦ᠂ ᠠ ᠶᠠᠯᠠᠭᠰᠠᠨ ᠵᠠᠮ ᠤᠨ ᠤᠷᠳᠤ ᠨᠢ 4 ᠳ᠋ᠦᠩᠰᠢᠭᠤᠷ ᠺᠢᠯ ᠣᠮᠧᠲᠷ ᠢ ᠳᠠ ᠪᠠᠵᠦ᠂ ᠭ ᠠᠯ ᠭᠠᠷ ᠠᠭ ᠠᠴᠠ ᠪᠠᠷᠤᠭ 83 0 ᠲᠥᠮᠡ ᠨ ᠺᠢᠯ ᠣᠮᠧᠲᠷ ᠵᠠᠢ ᠲᠠᠢ ᠪᠣᠯᠪᠠ ᠃ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ 83 00 00 0 ᠢ ᠰᠢᠨᠵᠢᠯ ᠡᠬᠦ ᠤᠠ ᠬᠭᠠᠨ ᠴᠢ ᠲᠣᠭ ᠠ ᠲᠡ ᠮᠲᠡᠭᠯ ᠡᠬᠦ ᠠ ᠷᠭ ᠠ ᠪᠠᠷ ᠢᠯ ᠡᠷᠭᠡᠢ ᠯᠡᠪᠡᠯ ( ) A. 8.3 × 10 6 B. 8. 3× 10 5 C. 83 × 10 5 D. 0. 83 × 10 6 3. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠣᠭ ᠨᠣᠪᠰᠢ ᠶᠢ ᠲᠥ ᠷᠥᠯ ᠬᠤᠪᠢᠶ ᠠᠭᠰᠠᠨ ᠲᠡ ᠮᠲᠡᠭ ᠦᠳ ᠲᠦ ᠂ ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠲᠦ ᠲᠡᠭᠰᠢ ᠬᠡᠮᠲᠠᠢ ᠮᠥᠷᠲᠡ ᠭᠡᠨ ᠪᠠᠰᠠ ᠲᠥᠪ ᠲᠦ ᠲᠡ ᠭᠰᠢ ᠬᠡᠮᠲᠠᠢ ᠵᠢᠷ ᠤᠭ ᠲᠥ ᠷᠰᠤ ᠪᠣᠯᠬᠤ ᠨᠢ ( ) A. B. C. D . 4. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠬᠡᠯᠡᠯᠭᠡ ᠨᠥᠬᠥ ᠳ ᠡᠴᠡ ᠵᠥᠪ ᠨᠢ ( ) A. ︽ᠬᠠᠩᠰᠢ ᠶᠢᠨ ᠡᠳᠤᠷ ᠲᠤ ᠪᠣᠷᠣᠭ ᠠᠨ ᠣ ᠷᠣᠬᠤ ︾ ᠨᠢ ᠵᠠ ᠢᠯᠠᠰᠢᠬᠦ ᠢ ᠶᠠᠪᠤ ᠳᠠᠯ B. ᠨᠢᠭᠡ ᠪᠥᠯᠥ ᠭ ᠳ᠋ᠧᠩ ᠫᠣᠣ ᠽᠢ ᠶᠢᠨ ᠬᠡᠷ ᠡᠭᠯᠡᠵᠦ ᠲᠠᠢᠯ ᠭᠦᠢᠴᠡ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭ ᠠ ᠶᠢ ᠣᠢᠯ ᠠᠭᠠᠬᠤ ᠶᠢᠨ ᠲᠥᠯᠥᠭᠡ᠂ ᠨᠠᠢᠲᠡᠮ ᠪᠠᠢᠴᠠᠭ ᠠᠬᠤ ᠠᠷᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ ᠪᠣᠯᠣᠨ ᠠ᠃ C. ᠨᠢᠭᠡ ᠪᠥ ᠯᠥᠭ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ 2,5,4 ,5, 6,7 ᠶᠢ ᠨ ᠮᠣᠳ᠋ ᠂ ᠳᠤᠮᠳ ᠠ ᠪ ᠠᠢᠷᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠪᠠ ᠳᠤᠮ ᠳᠠᠴᠢ ᠲᠣᠭ ᠠ ᠨᠢ ᠪᠦᠷ 5 D. ᠠ᠂ ᠡ ᠬᠣᠶ ᠠᠷ ᠠ ᠩᠭᠢ ᠶᠢᠨ ᠲᠠᠮᠢᠷ ᠴᠢᠨ ᠤ ᠪ ᠡᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠤᠨ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠶᠢᠨ ᠰᠠᠷᠨᠢᠴᠡ ᠨᠢ ᠲᠤᠰ ᠪ ᠦᠷᠢ 𝑆 ᠠ 2 = 0.02 , 𝑆 ᠡ 2 = 0.01 ᠂ ᠲᠡᠭᠡᠪ ᠡᠯ ᠡ ᠠᠩᠭ ᠢ ᠶᠢᠨ ᠲᠠᠮ ᠢᠷᠴᠢᠨ ᠤ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠦᠨᠳᠤᠷ ᠨᠢ ᠨᠡᠯᠢ ᠶᠡᠳ ᠳᠡᠭ ᠰᠢᠷᠡᠮ ᠡᠭ 5. ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠪᠣᠳᠣ ᠯᠭ ᠠ ᠠᠴ ᠠ ᠵᠥ ᠪ ᠨᠢ ( ) A. 𝑎− (𝑏 + 𝑐) = 𝑎− 𝑏+ 𝑐 B. 𝑎 3 + 𝑎 3 = 2𝑎 6 C. (𝑥 + 1) 2= 𝑥 2 + 1 D. 2𝑎 2⋅ (− 2𝑎 𝑏 2 ) 3 = −16 𝑎 5 𝑏 6 6. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ AB //CD ᠂ ᠴᠡᠭ E ᠨᠢ ᠬᠡ ᠷᠴᠢᠮ BC ᠳᠡ ᠭᠡᠷ ᠡ ᠪᠠᠢᠵᠤ᠂ CD = CE ᠂ ᠬᠡ ᠷᠪᠡ ∠AB C= 30° ᠪᠣᠯ ∠ D ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠨᠢ ( ) A. 85° B. 75° C. 65° D . 30° 7. ᠪᠣᠳᠠᠳᠤ ᠲᠣᠭ ᠠ 𝑎 ,𝑏 ,𝑐 ᠶᠢᠨ ᠲᠣᠭ ᠠᠨ ᠤ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠬᠠᠷ ᠠᠭᠠᠯᠵ ᠠᠭᠰᠠᠨ ᠴᠡᠭ ᠦᠨ ᠪᠠᠢᠷ ᠢ ᠶᠢ ᠳᠣᠣᠷ ᠠᠬᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠪᠡ᠃ ᠬᠡᠷ ᠪᠡ 𝑎+ 𝑏= 0 ᠪ ᠣᠯ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠳ᠋ᠦᠩ ᠡᠴᠡ ᠵᠥᠪ ᠨᠢ ( ) A. |𝑎| |𝑐| B . 𝑎 + 𝑐 0 ᠦᠶ ᠡᠰ ᠲᠦ᠂ 𝑥 ᠦᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠠᠪᠬᠤ ᠬᠡᠪᠴᠢᠶ ᠡ ᠨᠢ 0 𝑥 2 13. ᠨᠢᠭᠡᠨ ᠭᠧᠣᠮᠧᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠭᠤᠷ ᠪᠠᠨ ᠬᠠᠷᠠᠭ ᠠᠨ ᠵᠢᠷᠤᠭ ᠢ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠪᠣᠯ ᠡᠨᠡ ᠬᠦ ᠭᠧᠣᠮᠧᠲᠷ ᠢ ᠶᠢᠨ ᠪᠡᠶ ᠡ ᠶᠢᠨ ᠬᠠᠵᠠᠭ ᠤ ᠭᠠᠳᠠ ᠷᠭᠤ ᠶᠢ ᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ( ) A. 24 𝜋𝑐 𝑚 2 B. 48 𝜋𝑐 𝑚 2 C. 96 𝜋𝑐 𝑚 2 D . 36 𝜋𝑐 𝑚 2 14. ᠠ ᠂ ᠡ ᠬᠣ ᠶᠠᠷ ᠬᠦᠮᠦ ᠨ ᠡ ᠭᠢᠯᠡᠯ ᠲᠡ ᠶᠢᠨ ᠴᠡᠭ ᠠᠳᠠᠯᠢ᠂ ᠲᠡ ᠭᠦᠰᠭᠡᠯ ᠦᠨ ᠴᠡᠭ ᠠᠳᠠᠯᠢ᠂ ᠴᠢᠭᠯᠡᠯ ᠨᠢ ᠠ ᠳᠠᠯᠢ᠂ ᠤᠷ ᠳᠤ ᠨᠢ 400 ᠮᠧ ᠲᠷ ᠣᠯᠬ ᠤ ᠠ ᠰᠬᠠᠨ ᠬᠦᠶᠦ ᠳᠡᠯ ᠦᠨ ᠵᠠᠮ ᠳᠡᠭᠡᠷ ᠡ ᠵᠢᠭᠳᠡ ᠬᠤᠷᠳᠤᠴᠡ ᠪᠠ ᠷ ᠬᠦᠶᠦ ᠵᠦ᠂ ᠤᠷᠢᠳᠠ ᠪᠠᠷ ᠲᠡᠭᠦᠰ ᠭᠡᠯ ᠦ ᠨ ᠴᠡ ᠭ ᠲᠦ ᠬᠦᠷᠦ ᠭᠰᠡᠨ ᠬᠦ ᠮᠦᠨ ᠨᠢ ᠲᠤᠰ ᠭᠠᠵᠠᠷ ᠲᠤ ᠪᠠᠨ ᠠ ᠮᠠᠷᠠᠨ ᠠ᠃ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ᠠ ᠨᠢ ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ 3 ᠰᠧᠺᠦᠨᠲ ᠡᠮᠤᠨ ᠡ ᠡᠬ ᠢᠯᠡᠵ ᠠᠢ᠂ ᠬᠦᠶ ᠦᠬᠦ ᠶᠠ ᠪᠤᠴᠠ ᠳ᠋ᠤ᠂ ᠠ ᠡ ᠬᠣᠶ ᠠᠷ ᠬᠦᠮᠦᠨ ᠦ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠤᠬ ᠢ ᠵᠠ ᠢ 𝑦 (ᠮᠧ ᠲᠷ) ᠪᠣᠯ ᠣᠨ ᠡ ᠬᠦ ᠮᠦᠨ ᠡᠬᠢᠯ ᠡᠬᠦ ᠬᠤᠭ ᠤᠴᠠᠭ ᠠ 𝑥 ( ᠰᠧᠺᠦᠨ ᠲ) ᠤᠨ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠹᠦᠩᠺᠼ ᠦᠨ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠤᠨ ᠵᠢᠷᠤᠮᠠᠭ ᠢ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠭᠰᠡᠨ ᠪᠣᠯ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠦᠬᠦᠯᠡᠯ ᠦ ᠳ ᠡ ᠴᠡ ᠵᠥᠪ ᠦᠬ ᠦᠯᠡᠯ ᠦᠨ ᠪᠣᠳᠠᠭ ᠠᠯᠢ ᠲᠣᠭ ᠠ ᠨᠢ ( ) ① ᠡ ᠶᠢ ᠨ ᠬᠤᠷᠳᠤᠴᠠ ᠨᠢ 5 ᠮᠧ ᠲᠷ/ ᠰ ᠧᠺᠦᠨ ᠲ ② ᠡᠬᠢᠯ ᠡᠯᠡᠲᠡ ᠶᠢᠨ ᠴᠡᠭ ᠡᠴᠡ ᠵᠠᠢᠳᠠᠭᠰᠠᠨ ᠤ ᠳᠠᠷᠠᠭ ᠠ᠂ ᠠ ᠂ ᠡ ᠬᠣᠶ ᠠᠷ ᠬᠦ ᠮᠦᠨ ᠨᠢᠭᠡᠳᠤᠭᠡᠷ ᠤᠳᠠ ᠭ ᠠ ᠤᠴᠠ ᠷᠠᠬᠤ ᠦᠶ ᠡᠰ᠂ ᠡᠬᠢ ᠯᠡᠯᠲ ᠡ ᠶᠢ ᠨ ᠴᠡᠭ ᠡᠴᠡ 12 ᠮᠧᠲᠷ ᠵᠠᠢ ᠲᠠᠢ᠂ ③ ᠠ᠂ ᠡ ᠬᠣᠶ ᠠᠷ ᠬᠦᠮᠦ ᠨ ᠦ ᠬᠣᠭᠣᠷ ᠣᠨᠳᠣᠬ ᠢ ᠵᠠ ᠢ ᠨᠢ 32 ᠮᠧ ᠲᠷ ᠡᠴᠡ ᠬᠡᠲᠤᠷᠡ ᠬᠦ ᠦ ᠶᠡᠰ ᠲᠦ ᠬᠤᠭᠤᠴᠠᠭ ᠠᠨ ᠤ ᠬᠡᠪᠴᠢᠶ ᠡ ᠨᠢ 44 𝑥 89 ④ ᠡ ᠨᠢ ᠲᠡᠭᠦᠰ ᠭᠡᠯ ᠦ ᠨ ᠴᠡᠭ ᠲᠦ ᠬᠦᠷᠬ ᠦ ᠦᠶ ᠡᠰ ᠲᠦ᠂ ᠠ ᠨᠢ ᠲᠡᠭᠦᠰ ᠭᠡᠯ ᠦ ᠨ ᠴᠡ ᠭ ᠡᠴᠡ ᠪᠠᠰᠠ 68 ᠮᠧᠲᠷ ᠵᠠᠢ ᠲᠠᠢ A. 4 B. 3 C. 2 D .1 ᠬᠣᠶ ᠠᠷ᠂ ᠬᠣᠭᠣᠰᠤ ᠨ ᠳ᠋ ᠤ ᠲᠠ ᠭᠯᠠᠬᠤ ᠰᠡᠲᠤᠪ ᠃ ( ᠰᠡᠲᠤᠪ ᠪᠦᠷᠢ 3 ᠬᠤᠪᠢ᠂ ᠨᠠᠢᠲᠡ 12 ᠬᠤᠪᠢ) 𝑥 𝑦. . −1 3 3 −1 𝑚3 . . 15. ᠹᠦᠩᠺᠼ 𝑦 = √𝑥 +1 2𝑥 −1 ᠳ᠋ᠦ᠂ ᠬᠤᠪᠢ ᠰᠬᠤ ᠨ 𝑥 ᠤᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡᠪ ᠴᠢᠶ ᠡ ( ) 16. ᠨᠢᠭᠡᠨ ᠴᠠᠰᠤ ᠨ ᠳ᠋ ᠤ ᠭᠣᠯ ᠭᠤᠯᠬ ᠤ ᠲᠠ ᠯᠠᠪᠠᠢ ᠶᠢ ᠤᠭᠲᠠᠷ ᠭᠤᠢ ᠠ ᠴᠠ ᠰᠡᠭᠦᠳᠡ ᠷᠯᠡᠬᠦ ᠨᠢᠰ ᠭᠡᠯ ᠢ ᠬᠡᠷᠡᠭᠯᠡ ᠨ ᠭᠤᠯ ᠭᠤᠬ ᠤ ᠵᠠ ᠮ ᠤᠨ ᠤᠷ ᠳᠤ ᠶᠢ ᠬᠡᠮᠵᠢᠪᠡ᠃ ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠤᠭᠲᠠᠷ ᠭᠤᠢ ᠠ ᠴᠠ ᠰᠡᠭᠦᠳᠡ ᠷᠯᠡᠬᠦ ᠨᠢ ᠰᠭᠡᠯ ᠳ᠋ᠠᠬ ᠢ ᠳᠤᠷ ᠠᠨᠳᠠᠭ ᠤᠷ 𝐶 ᠠᠴᠠ ᠬᠡᠪᠲᠡ ᠭᠡ ᠭ ᠤᠯᠭᠤ ᠬᠤ ᠵᠠᠮ ᠤ ᠨ ᠨᠢ ᠭᠡᠨ ᠦᠵᠦᠭ ᠦᠷ ᠦ ᠨ ᠴᠡ ᠭ 𝐴 ᠳ᠋ᠠᠬᠢ ᠰᠢᠷᠭ ᠤᠭᠰ ᠠᠨ ᠥᠨᠴᠥ ᠭ 50° ᠂ ᠨᠥᠬᠥ ᠭᠡ ᠨᠢᠭᠡ ᠥᠵᠥᠭ ᠦᠷ ᠦ ᠨ ᠴᠡ ᠭ 𝐵 ᠳ᠋ᠠᠬᠢ ᠰᠢᠷᠭ ᠤᠭᠰ ᠠᠨ ᠥᠨᠴ ᠥᠭ 45° ᠪᠣᠯ ᠬᠤ ᠶᠢ ᠬᠡ ᠮᠵᠢᠨ ᠣ ᠯᠵᠠᠢ᠂ ᠬᠡᠷᠪᠡ ᠤᠭᠷ ᠠᠷᠭᠤᠢ ᠠᠴ ᠠ ᠰ ᠡᠭᠦᠳᠡ ᠷᠯᠡᠬᠦ ᠨᠢᠰᠭᠡᠯ ᠦᠨ ᠳᠤᠷᠠ ᠨᠳᠠᠭ ᠤᠷ 𝐶 ᠳ᠋ᠠᠬ ᠢ ᠦᠨᠳᠤᠷ 𝐶𝐷 ᠨᠢ 238 ᠮᠧᠲᠷ ᠂ ᠴᠡᠭ 𝐴, 𝐷 , 𝐵 ᠨᠢ ᠢᠵᠢ ᠯᠢ ᠨᠢᠭᠡᠨ ᠰᠢᠯᠤᠭᠤᠨ ᠳᠡ ᠭᠡᠷ ᠡ ᠪᠠᠢᠪ ᠠᠯ ᠭᠤᠯ ᠭᠤᠬ ᠤ ᠵᠠ ᠮ 𝐴𝐵 ᠶᠢᠨ ᠤᠷ ᠳᠤ ᠨᠢ ( ) 𝑚 ( ᠳ᠋ᠦ ᠩ ᠳ᠋ ᠦ ᠪᠥᠬᠥ ᠯᠢ ᠲᠣᠭ ᠠ ᠶᠢ ᠦᠯ ᠡᠳᠡᠭᠡ᠂ ᠯᠠᠪ ᠯᠠᠬᠤ ᠲᠣᠭ ᠠ ᠪᠠᠷᠢᠮᠲᠠ 𝑠𝑖𝑛 50° ≈ 0.77 , 𝑐𝑜𝑠 50° ≈ 0.64 , 𝑡𝑎𝑛 50° ≈ 1.19 17. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠠᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 𝑎 ᠪᠣᠯᠬᠤ ᠵᠥᠪ ᠵᠢᠷᠭ ᠤᠭᠠᠨ ᠥᠨᠴᠥ ᠭᠳᠥ ᠱᠤᠷᠠᠭ ᠬᠠᠳᠠ ᠭᠠᠰᠤ ᠶᠢ ᠮᠤ ᠰᠬᠢᠬ ᠤ ᠳ᠋ᠤ᠂ ᠮᠤᠰᠬᠢᠭ ᠤᠷ ᠤ ᠨ ᠠ ᠩᠭᠠᠢᠭᠰᠠᠨ ᠠᠩᠭᠠ ᠢᠯᠲᠠ ᠪᠣᠯ 20 𝑚𝑚 ᠪᠣᠯ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤ ᠷᠳᠤ 𝑎= ( ) 𝑚𝑚 ᠃ 18. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠲᠡᠭᠰᠢ ᠲᠥᠷᠪ ᠡᠯᠵᠢ ᠨ 𝐴𝐵𝐶𝐷 ᠶᠢᠨ ᠲᠠ ᠯ ᠠ ᠶᠢᠨ ᠤᠷᠳᠤ ᠨᠢ 2√ 5᠂ ᠴᠡᠭ 𝐸 ᠪᠣᠯ 𝐵𝐶 ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡ ᠭ᠂ 𝐴𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ ᠳᠠᠰᠢᠭᠤ ᠪᠴᠢ 𝐵𝐷 ᠯᠤᠭ ᠠ ᠴᠡᠭ 𝐹 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠵᠤ᠂ 𝐷𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ᠂ 𝐶𝐹 ᠯᠤᠭ ᠠ ᠴᠡᠭ ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠵᠤ᠂ 𝐴𝐻 ᠶᠢ ᠬᠣᠯ ᠪᠣᠨ ᠠ᠃ ① 𝐶𝐹 ⊥ 𝐷𝐸 ② 𝐶𝐻𝐻𝐹 = 23 ③ 𝐺𝐻 = 23 √ 5 ④ 𝐴𝐷 = 𝐴𝐻 ᠂ ᠡᠮᠤᠨᠡᠬ ᠢ ᠦᠬᠦᠯᠡᠯ ᠨᠣᠬᠥᠳ ᠠᠴᠠ ᠵᠥ ᠪ ᠦᠬᠦᠯᠡᠯ ᠦᠨ ᠳ᠋ᠤᠭᠠᠷ ᠲᠡ ᠮᠲᠡᠭ ᠨᠢ ( ) ᠭᠤᠷᠪ ᠠ᠂ ᠪᠣᠳᠣ ᠵᠤ ᠬᠠᠷ ᠢᠭᠤᠯᠬᠤ ᠰᠡ ᠲᠤᠪ᠃ ( ᠨᠠ ᠢᠲᠡ 96 ᠬᠤᠪᠢ) 19. (1 0 ᠬᠤᠪᠢ ) ᠤᠷ ᠢᠳᠠᠪᠠ ᠷ ᠬᠢᠯ ᠪᠠᠷᠴᠢ ᠯᠠᠵᠤ ᠳᠠ ᠷᠠᠭ ᠠ ᠨᠢ ᠬᠡ ᠮᠵᠢᠭ ᠳᠡᠯ ᠢ ᠨᠢ ᠣᠯ᠃ 𝑚 −3 𝑚 −2 ÷ (𝑚 + 2− 5𝑚−2 )᠂ ᠡᠭᠦᠨ ᠳ᠋ᠦ 𝑚 = ( 13 ) −1 + (2 − 𝑥) 0+ √8 − |7| 20. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ 𝑅𝑡 ∆𝐴𝐵𝐶 ᠳ᠋ᠦ᠂ ∠𝐴𝐶𝐵 = 90° ᠂ ᠴᠡ ᠭ 𝐷 ᠪᠣᠯ ᠵᠢᠰᠢ ᠬᠦᠦ ᠲᠠᠯ ᠠ 𝐴𝐵 ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡᠨ ᠴᠡᠭ ᠪᠥᠭᠡᠳ 𝐴𝐶 = 𝐴𝐷 ᠃ (1) ∠𝐵𝐴𝐶 ᠶᠢᠨ ᠲᠠ ᠯᠠᠯᠠᠨ ᠬᠤᠪᠢ ᠶᠠᠭᠴᠢ ᠱᠤᠭᠤᠮ ᠢ ᠪ ᠠᠢᠭ ᠤᠯᠵᠤ᠂ 𝐵𝐶 ᠯᠤᠭ ᠠ ᠴᠡ ᠭ 𝐸 ᠳ᠋ᠤ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠭᠤᠯᠤ ᠨ ᠠ᠃ ( ᠱᠠᠭ ᠠᠷᠳᠠᠯ ᠭ ᠠ᠄ ᠤᠤ ᠢ ᠭᠣᠷᠲᠢᠭ ᠢᠶᠠᠷ ᠵᠢᠷ ᠤᠭ ᠪᠠᠢᠭ ᠤᠯᠤᠨ ᠠ᠂ ᠪᠠᠢᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠠᠷᠭ ᠠ ᠪᠠᠨ ᠪᠢᠴᠢᠬᠦ ᠦᠭᠡᠢ ᠂ ᠪᠠᠢᠭ ᠤᠯᠤᠭᠰᠠᠨ ᠤᠯᠠ ᠮᠥᠷ ᠢᠶᠡ ᠨ ᠦᠯ ᠡᠳᠡᠭᠡ ) (2) ᠳᠡᠭᠡᠷ ᠡᠬᠢ (1) ᠶ ᠢᠨ ᠨᠥᠬᠥ ᠴᠡᠯ ᠳᠣᠣᠷ ᠠ 𝐷𝐸 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠃ ᠪᠠᠳᠤᠯ ᠠᠭᠳᠠ ᠬᠤ ᠨᠢ ᠄ 𝐷𝐸 ⊥ 𝐴𝐵 21. ᠨᠢᠭᠡᠨ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠶᠢᠰᠤᠳ ᠤᠭᠡᠷ ᠣᠨ ᠳ᠋ ᠤ 1 2 ᠠ ᠩᠭᠢ ᠪᠣᠢ ᠠᠩᠭᠢ ᠪᠦᠷᠢ 50 ᠰᠤᠷᠤᠭᠴᠢ ᠲᠠᠢ᠃ ᠲᠤᠰ ᠰᠤᠷᠭ ᠠᠭᠤᠯ ᠢ ᠶᠢᠨ ᠶᠢᠰᠤ ᠳᠤᠭᠡᠷ ᠣᠨ ᠤ ᠰ ᠤᠷᠤᠭᠴᠢᠳ ᠤ ᠨ ᠳᠤᠮ ᠳᠠᠴᠢ ᠪᠠᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤᠨᠳ ᠠᠬᠤ ᠴᠠᠭ ᠢ ᠪᠠᠢᠴᠠᠭ ᠠᠬᠤ ᠶᠢᠨ ᠲᠥ ᠯᠥᠭᠡ᠂ 12 ᠠᠩᠭᠢ ᠠᠴᠠ 50 ᠰᠤᠷᠤᠭ ᠴᠢ ᠶᠢ ᠰᠤᠭ ᠤᠯᠵᠤ᠂ ᠨᠢᠭᠡᠨ ᠶᠠᠩ ᠵᠤ ᠪ ᠣᠯᠭᠠᠨ ᠵᠠᠳᠠᠯᠤᠯ ᠲᠠ ᠬᠢᠬᠦ ᠪᠡᠷ ᠪᠡᠯ ᠡᠳᠭᠡᠵ ᠤ᠂ ᠪᠠᠰᠠ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠤ ᠳᠣᠭᠲᠠᠭ ᠠᠯ ᠭᠠᠷ ᠭᠠᠵᠠᠢ ᠄ ᠰ ᠤᠷᠤᠭᠴᠢ ᠨᠢ ᠭᠡᠪᠤᠷᠢ ᠶᠢᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤᠨᠳᠠᠬᠤ ᠴᠠᠭ ᠢ 𝑡 (ᠨᠢᠭᠡᠴᠢ ᠄ ᠴᠠ ᠭ ) ᠭᠡᠵᠤ ᠵᠢᠱᠢ ᠵᠤ᠂ ᠴᠤᠭᠯ ᠠᠭᠤᠯᠤᠨ ᠤᠯᠣᠭᠰ ᠠᠨ ᠰᠤᠷᠤᠭ ᠴᠢᠳ ᠤᠨ ᠳᠤᠮ ᠳᠠᠴᠢ ᠪᠠ ᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤᠨᠳ ᠠᠬᠤ ᠴᠠᠭ ᠢ 𝑡≤ 6,6 𝑡 8, 𝑡≥ 8 ᠭᠡᠵ ᠤ ᠭᠤᠷᠪ ᠠᠨ ᠲᠥᠷᠥᠯ ᠬᠤᠪᠢ ᠶᠠᠵᠤ ᠵᠠᠳᠠᠯᠤᠯ ᠲᠠ ᠬᠢᠨ ᠡ᠃ (1) ᠳᠠᠷ ᠠᠭᠠᠬ ᠢ ᠰ ᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠬᠤ ᠠᠷᠭ ᠠ ᠨᠤᠭ ᠤᠳ ᠲᠤ ᠲᠥ ᠯᠥᠭᠡᠯ ᠡᠬᠦ ᠴᠢᠨ ᠠᠷ ᠲ ᠠᠢ ᠨᠢ A. ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠ ᠷ ᠨᠢᠭᠡ ᠠᠩᠭᠢ ᠶᠢᠨ ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠ B. 12 ᠠᠩᠭ ᠢ ᠠᠴᠠ ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ 50 ᠰᠤᠷᠤᠭᠴᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠ C. ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠᠷ 50 ᠡᠷᠡᠭᠲᠡᠢ ᠰᠤᠷ ᠤᠭᠴ ᠢ ᠰᠤᠭᠤ ᠯᠤᠨ ᠠ D. ᠳᠠᠰᠢᠷᠠ ᠮ ᠢᠶᠠ ᠷ 50 ᠬᠡᠤᠬᠡᠨ ᠰᠤᠷᠤ ᠭᠴᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠ (2) ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠲᠥᠯᠥᠭᠡᠯ ᠡᠬᠦ ᠴᠢᠨᠠᠷ ᠲᠠ ᠢ ᠰ ᠤᠭᠤᠯ ᠬᠤ ᠠᠷᠭ ᠠ ᠪᠠᠷ 50 ᠰᠤᠷᠤᠭᠴᠢ ᠶᠢ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠤᠪᠠᠯ᠂ ᠳᠤᠮ ᠳᠠᠴᠢ ᠪ ᠠᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶ ᠢᠨ ᠤᠨᠳ ᠠᠬᠤ ᠴᠠᠭ ᠤᠨ ᠲᠣᠭ ᠠ ᠪᠠᠷ ᠢᠮᠲᠠ ᠨᠢ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠮᠡᠲᠤ ᠄ ① ᠡᠨ ᠡ ᠪ ᠥᠯᠥ ᠭ ᠲᠣᠭ ᠠ ᠪ ᠠᠷᠢᠮᠲᠠ ᠶ ᠢᠨ ᠮᠤᠳ᠋ ᠪᠠ ᠳᠤᠮᠳᠠ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠲᠤᠰ ᠲᠤᠰ ( ) ( ) ② ᠶᠢᠰ ᠤᠳᠤᠭᠡᠷ ᠣᠨ ᠤ ᠰᠤᠷᠤᠭᠴᠢᠳ ᠤᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤ ᠳᠠᠬᠤ ᠴᠠᠭ 𝑡 ≥ 8 ᠪᠣᠯᠬᠤ ᠬᠦᠮᠦ ᠨ ᠦ ᠲᠣᠭ ᠠ ᠬᠡ ᠳᠤᠢ ᠪᠠᠢᠬ ᠤ ᠶᠢ ᠪᠠᠷ ᠤᠭᠴ ᠠᠭᠠᠯᠠ ᠃ (3) ᠶᠠᠩ ᠵᠤ ᠳ᠋ᠠᠬ ᠢ ᠰ ᠤᠷᠤᠭᠴᠢᠳ ᠤᠨ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ ᠡᠳᠤᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤᠨᠳ ᠠᠬᠤ ᠴᠠᠭ ᠨᠢ 𝑡 ≤ 6 ᠪᠣᠯᠬᠤ ᠰᠤᠷᠤᠭᠴᠢ ᠠᠴᠠ ᠳᠠ ᠰᠢᠷᠠᠮ ᠢᠶᠠᠷ 2 ᠬᠦ ᠮᠦᠨ ᠢ ᠰᠤᠭᠤ ᠯᠵᠤ᠂ ᠮᠤᠳᠤ ᠯᠵᠢᠨ ᠵᠢᠷ ᠤᠭ ᠪᠠᠢᠭ ᠤᠯᠬᠤ ᠪᠤᠶᠤ ᠲᠣᠭ ᠠᠴᠢᠨ ᠵᠢᠭᠰᠠᠭ ᠠᠬᠤ ᠠᠷᠭ ᠠ ᠶᠢ ᠬᠡᠷᠡᠭᠯᠡᠵᠦ᠂ ᠰᠤᠭᠤᠯ ᠤᠨ ᠠᠪᠤᠭ ᠰᠠᠨ 2 ᠬᠦᠮᠦ ᠨ ᠦ ᠳᠤᠮᠳᠠᠴᠢ ᠪ ᠠᠷ ᠡᠳᠤ ᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠤᠨᠳ ᠠᠬᠤ ᠴᠠᠭ ᠨᠢ ᠴᠥ ᠮ 6 ᠴᠠᠭ ᠪᠣᠯᠬᠤ ᠳᠣᠬᠢᠶ ᠠᠯᠳᠤᠮᠵᠢ ᠶᠢ ᠣᠯ᠃ 22. (1 2ᠬᠤᠪᠢ) ᠰ ᠢᠯᠢᠳᠡ ᠭ ᠤᠯᠠᠮ ᠵᠢᠯᠠᠯᠲ ᠤ ᠰ ᠤᠶᠤᠯ ᠢ ᠤ ᠯᠠᠮᠵᠢ ᠯᠠᠬᠤ ᠶᠢᠨ ᠲᠥ ᠯᠥᠭᠡ᠂ ᠨᠢ ᠭᠡᠨ ᠭᠠᠵᠠᠷ ᠤᠨ ᠵᠠᠯ ᠠᠭᠤ ᠪᠠᠭᠠᠴᠤᠳ ᠤᠨ ᠬᠥᠳ ᠡᠯᠭᠡᠭᠡᠨ ᠦ ᠲ ᠥᠪ᠂ ᠤᠳᠠ ᠭ ᠠ ᠬᠤᠪᠢ ᠶᠠᠨ ᠳᠥᠷᠪ ᠡᠨ ᠠᠯᠳᠠᠷ ᠲᠤ ᠵᠣᠬᠢᠶᠠᠯ ︽ᠪᠠᠷ ᠠᠭᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡ ᠮᠲᠡᠭᠯᠡᠯ ︾᠂ ︽ᠱᠦ ᠢ ᠬᠤ ᠶᠢᠨ ᠱᠠᠰᠲᠢᠷ ︾ ︽ ᠭᠤ ᠷᠪᠠᠨ ᠤᠯᠤᠰ ᠤᠨ ᠦᠯ ᠢᠭᠡᠷ ︾᠂ ︽ᠤᠯ ᠠᠭᠠᠨ ᠠᠰᠠᠷ ᠤ ᠨ ᠵᠡ ᠭᠦᠳᠦ ︾ ᠶᠢ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠ ᠪᠬᠤ ᠪ ᠠᠷ ᠲᠥ ᠯᠥᠪᠯᠡᠵ ᠤ ᠪ ᠣᠢ᠃ ᠨᠢᠭᠡᠳ ᠤᠭᠡᠷ ᠤᠳᠠ ᠭ ᠠ ︽ᠪᠠᠷ ᠠᠭᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡᠮᠲᠡᠭᠯᠡᠯ ︾ ᠢ 50 ᠲᠡ ᠪᠳᠡᠷ ᠂ ︽ ᠱᠦᠢ ᠬᠤ ᠶᠢ ᠨ ᠱᠠᠰᠲᠢᠷ ︾ ᠢ 6 0 ᠲᠡ ᠪᠲᠡᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠ ᠪᠴᠤ ᠨᠠ ᠢᠲᠡ 6 60 0 ᠲᠥᠬᠥ ᠷᠢᠭ ᠵᠠᠷᠤᠴ ᠠᠭᠤᠯ ᠪᠠᠯ᠃ ᠬᠣ ᠶᠠᠳᠤᠭᠠ ᠷ ᠤᠳᠠ ᠭ ᠠ ︽ᠪᠠᠷ ᠠᠭᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡ ᠮᠲᠡᠭᠯᠡᠯ ︾ ᠢ 40 ᠲᠡᠪᠲᠡ ᠷ᠂ ︽ ᠰᠦᠢ ᠬᠤ ᠶᠢᠨ ᠱᠠᠰᠲ ᠢᠷ︾ ᠢ 3 0 ᠲᠡ ᠪᠲᠡᠷ ᠬᠤᠯ ᠳᠠᠳᠤᠨ ᠠᠪᠴᠤ᠂ ᠨᠠᠢᠲᠡ 42 00 ᠲᠥᠬᠥ ᠷᠢᠭ ᠵᠠᠷᠤᠴ ᠠᠭᠤᠯ ᠪᠠ᠃ (1) ︽ᠪᠠᠷ ᠠᠭᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡᠮᠲᠡᠭᠯᠡᠯ ︾ ᠪᠠ ︽ᠱᠦ ᠢ ᠬᠤ ᠶᠢᠨ ᠱᠠᠰᠲᠢᠷ ︾ ᠤᠨ ᠲᠡ ᠪᠲᠡᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦᠨ ᠡ ᠨᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠬᠡᠳ ᠤᠢ ᠲᠥᠬᠥ ᠷᠢᠭ ᠪᠣᠯᠬᠤ ᠶᠢ ᠣᠯ᠃ (2) ᠵᠠᠯᠠᠭ ᠤ ᠪ ᠠᠭᠠᠴᠤᠳ ᠤᠨ ᠬᠥᠳᠡ ᠯᠭᠡᠭᠡᠨ ᠦ ᠲᠥᠪ ᠨᠢ ᠡᠳᠡ ᠭᠡᠷ ᠳᠥᠷᠪ ᠡᠨ ᠵᠦᠢᠯ ᠦ ᠨ ᠨᠠ ᠪᠴᠢᠨ ᠣ ᠮᠢᠬ ᠠ ᠶᠢ ᠳᠠᠬᠢᠨ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠬᠤ ᠳ᠋ ᠤ ᠳᠣᠭᠲᠠ ᠵᠤ᠂ ᠶᠡᠷ ᠤᠩᠬᠡᠢ ᠵᠠᠷᠤᠳᠠᠯ ᠨᠢ 3 20 00 ᠲ ᠥᠬᠥ ᠷᠢᠭ ᠡᠴᠡ ᠬᠡᠲᠤᠷᠡ ᠬᠦ ᠦᠭᠡᠢ᠂ ᠬᠡᠷᠪᠡ ︽ᠪ ᠠᠷᠠᠭ ᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡ ᠮᠲᠡᠭᠯᠡᠯ ︾ ᠨᠢ ︽ᠭᠤᠷᠪ ᠠᠨ ᠤᠯ ᠤᠰ ᠤ ᠨ ᠦ ᠯᠢᠭᠡᠷ ︾ ᠡᠴᠡ ᠳᠡᠪᠲᠡ ᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦᠨᠡᠨ ᠨᠢ ᠢᠯᠡᠭ ᠦᠦ 10 ᠲᠥᠬᠥ ᠷᠢᠭ᠂ ︽ ᠱᠦᠢ ᠬᠤ ᠶᠢᠨ ᠱᠠᠰᠲᠢᠷ ︾ ᠨᠢ ︽ᠤ ᠯᠠᠭᠠᠨ ᠠᠰᠠᠷ ᠤᠨ ᠵᠡᠭᠦᠳᠦ ︾ ᠡᠴ ᠡ ᠳᠡ ᠪᠲᠡᠷ ᠪᠦᠷᠢ ᠶᠢ ᠨ ᠦᠨᠡᠨ ᠨᠢ ᠳᠤ ᠲᠠᠭᠤ 10 ᠲᠥᠭ ᠥᠷᠢᠭ ᠂ ᠤᠷᠢᠳ ᠬᠣᠵ ᠢᠳ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪᠤᠭ ᠰᠠᠨ ᠳᠥᠷᠪ ᠡᠨ ᠠ ᠯᠳᠠᠷ ᠳᠤ ᠵᠣᠬᠢᠶᠠᠯ ᠢᠶᠠᠨ ᠶᠠᠭ ᠰᠠᠢᠬᠠᠨ ᠢᠵ ᠢ ( ᠳᠥᠷᠪ ᠡᠨ ᠠ ᠯᠳᠠᠷ ᠲᠤ ᠵᠣᠬᠢ ᠶᠠᠯ ᠨᠢ ᠲᠤᠰ ᠪᠦᠷᠢ ᠨᠢᠭᠡ ᠲᠡ ᠪᠳᠡᠷ ᠪᠠᠢᠪ ᠠᠯ ᠨᠢ ᠭᠡ ᠵᠠᠢ ᠶᠢ ᠪᠦᠷᠢᠯᠳ ᠥᠬᠦᠯ ᠤᠨ ᠡ ) ᠪᠦᠷᠢᠯᠳᠥ ᠬᠦᠯᠪᠡ ᠯ᠂ ᠡᠨᠡ ᠤᠳᠠ ᠭ ᠠ ︽ᠪᠠᠷ ᠠᠭᠤᠨᠰᠢ ᠵᠣᠷᠴᠢᠭᠰ ᠠᠨ ᠲᠡ ᠮᠲᠡᠭᠯᠡᠯ ︾ ᠢ ᠬᠠᠮ ᠤᠭ ᠠᠷᠪ ᠢᠨ ᠳ᠋ᠤ ᠪ ᠠᠨ ᠬᠡ ᠳᠤᠦᠢ ᠲᠡᠪᠳ ᠡᠷ ᠬᠤᠳᠠᠯ ᠳᠤᠨ ᠠᠪ ᠬᠤ ᠪ ᠣᠢ︖ 23. (1 2 ᠬᠤᠪᠢ) ᠣᠩᠰᠢᠨ ᠣ ᠢᠯᠠᠭ ᠠᠬᠤ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤ ᠨ ᠰᠢᠰᠲᠧᠮ ᠳ᠋ᠦ᠂ ᠴᠡ ᠭ 𝑀 ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( 𝑥1 ,𝑦 1 )᠂ ᠴᠡᠭ 𝑁 ᠤ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (𝑥 2,𝑦 2 ) ᠪᠥᠭᠡᠳ 𝑥 1≠ 𝑥2 , 𝑦 1≠ 𝑦2 ᠂ ᠬᠡ ᠷᠪᠡ 𝑀 ,𝑁 ᠨᠢ ᠨᠢᠭᠡᠨ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢᠨ ᠬᠣᠶ ᠠᠷ ᠣᠷᠣᠢ ᠶᠢᠨ ᠴᠡ ᠭ ᠪ ᠥᠭᠡᠳ ᠲᠤᠰ ᠲᠡᠭᠰᠢ ᠥᠨᠴ ᠥᠭᠳᠥ ᠶᠢᠨ ᠲᠠ ᠯ ᠠ ᠨᠢ ᠪᠦᠷ ᠶᠠᠮᠠᠷ ᠨᠢᠭᠡᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠤᠨ ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠯᠦᠭᠡ ᠡᠭᠴᠡ ᠪᠣᠯ ᠲᠤᠰ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭᠳᠥ ᠶ ᠢ 𝑀 ,𝑁 ᠤ ︽ᠬᠠ ᠮᠢᠶ ᠠ ᠪᠥᠬᠥ ᠢ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥ ᠭᠳᠥ︾ ᠭᠡᠨ ᠡ᠃ (1) ᠦᠭ ᠭᠦᠭᠳᠡᠭ ᠰᠡᠨ ᠨᠢ ᠴᠡᠭ 𝐴 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( 2,0 ) ① ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ 𝐵 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( 4,4 ) ᠪᠣᠯ ᠴᠡᠭ 𝐴 ,𝐵 ᠶᠢᠨ ︽ᠬᠠᠮ ᠢᠶ ᠠ ᠪᠥᠬᠥ ᠢ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭᠳᠥ ︾ ᠶᠢᠨ ᠬᠦᠷ ᠢᠶᠡᠨ ᠤᠷᠳᠤ ᠨᠢ ( ) ② ᠬᠡᠷ ᠪᠡ ᠴᠡᠭ 𝐶 ᠨᠢ ᠰᠢᠯᠤᠭᠤᠨ 𝑥 = 4 ᠳ ᠡᠭᠡᠷ ᠡ ᠣᠷᠣ ᠰᠢᠬ ᠤ ᠪᠥ ᠭᠡᠳ ᠴᠡ ᠭ 𝐴 ,𝐶 ᠶᠢᠨ ︽ᠬᠠᠮ ᠢᠶ ᠠ ᠪᠥᠬᠦ ᠢ ᠲᠡ ᠭᠰᠢ ᠥᠨᠴᠥᠭᠳᠥ ︾ ᠨᠢ ᠲᠡᠭᠰᠢ ᠳᠥᠷᠪ ᠡᠵᠵᠢᠨ ᠪᠣᠯ ᠰᠢᠯ ᠤᠭᠤᠨ 𝐴𝐶 ᠶᠢᠨ ᠠ ᠨᠠᠯᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢ ᠣᠯ᠃ (2) ᠦ ᠭᠭᠦᠭᠳᠡ ᠭᠰᠡᠨ ᠨᠢ ᠴᠡᠭ 𝑃 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ ( 3, −4 )᠂ ᠴᠡᠭ 𝑄 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠨᠢ (6 ,− 2) ᠂ ᠬᠡ ᠷᠪᠡ ᠹᠦᠩᠺᠼ 𝑦 = 𝑘𝑥 ᠦ ᠨ ᠵᠢᠷ ᠤᠮᠠᠭ ᠪᠣᠯᠣᠨ ᠴᠡᠭ 𝑃 ᠂ 𝑄 ᠶᠢᠨ ︽ᠬᠠᠮ ᠢᠶ ᠠ ᠪᠥᠬᠦ ᠢ ᠲᠡᠭᠰᠢ ᠥᠨ ᠴᠥᠭᠳᠥ ︾ ᠯᠤ ᠭ ᠠ ᠬᠣᠶ ᠠᠷ ᠶᠡᠷ ᠤᠩᠬᠡᠢ ᠴᠡᠭ ᠲᠠᠢ ᠪᠣᠯ 𝑘 ᠤ ᠨ ᠬᠡ ᠮᠵᠢᠭᠳᠡᠯ ᠠ ᠪᠬᠤ ᠬᠡᠪᠴᠢᠶ ᠡ ᠶᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴᠢᠨ ᠭᠠᠷᠭ ᠠ᠃ 24. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠷᠤᠮᠪ 𝐴𝐵𝐶𝐷 ᠳ᠋ᠤ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ 𝐴𝐶 ,𝐵𝐷 ᠨᠢ ᠴᠡᠭ 𝑀 ᠯᠤ ᠭ ᠠ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠵᠤ᠂ ⊙ O ᠨᠢ ᠴᠡᠭ 𝐵 ,𝐶 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠵ ᠤ᠂ ᠳᠠᠰᠢᠭᠤᠪ ᠴᠢ 𝐵𝐷 ᠯᠤᠭ ᠠ ᠴᠡᠭ 𝐸 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠬᠤ ᠪᠥᠭᠡᠳ 𝐶𝐸̂ = 𝐵𝐸̂ , 𝑂𝐸 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ 𝐵𝐶 ᠯᠤᠭ ᠠ ᠴᠡᠭ 𝐹 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠨ ᠠ᠃ (1) 𝐴𝐵 ᠪᠣᠯ ᠣᠨ ⊙ O ᠶᠢᠨ ᠪᠠᠢᠷ ᠢ ᠶᠢᠨ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠢ ᠳᠤᠷ ᠰᠢᠨ ᠮᠠ ᠭᠠᠳᠯ ᠠᠵᠤ᠂ ᠪᠠᠰ ᠠ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠨᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ (2) ᠬᠡᠷᠪᠡ 𝐵𝐷 = 325 √5 , 𝑡𝑎𝑛 ∠𝐶𝐵 𝐷 = 12 ᠪ ᠣᠯ ⊙ O ᠤᠨ ᠷᠠᠳᠢᠤᠰ ᠢ ᠣᠯ᠃ 25. ᠵᠢᠷ ᠤᠭ ᠮᠡᠲᠤ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ 𝑦 = −𝑥 2+ 𝑏𝑥 + 𝑐 ᠨᠢ 𝑥 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠯᠤᠭᠡ 𝐴( −3 ,0) , 𝐵 (1 ,0) ᠬᠣ ᠶᠠᠷ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠᠵ ᠤ᠂ 𝑦 ᠲᠡᠩᠭᠡᠯ ᠢᠭ ᠯᠦᠭᠡ ᠴᠡ ᠭ 𝐶 ᠳ᠋ᠦ ᠣᠭᠲᠣ ᠯᠣᠯᠴᠠ ᠵᠤ᠂ ᠲᠡᠭᠰᠢ ᠬᠡᠮ ᠦᠨ ᠲᠡᠩᠭᠡᠯ ᠢᠭ 𝑙 ᠦᠨ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠯᠦᠭᠡ ᠣ ᠭᠲᠣᠯ ᠣᠯᠴᠠ ᠯ ᠤᠨ ᠴᠡ ᠭ ᠨᠢ 𝐹᠂ ᠰᠢᠯᠤᠭᠤᠨ 𝑚 ∕∕ 𝐴𝐶 ᠂ ᠴᠡᠭ 𝐸 ᠪᠣᠯ ᠰᠢᠯᠤᠭᠤ ᠨ 𝐴𝐶 ᠶᠢᠨ ᠳᠡ ᠭᠡᠷ ᠡ ᠲᠠᠯ ᠠ ᠳ᠋ᠠᠬ ᠢ ᠬᠠᠶᠠ ᠯᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤᠮ ᠳᠡ ᠭᠡᠷᠡᠬ ᠢ ᠨᠢ ᠭᠡ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡᠭ ᠂ ᠴᠡᠭ 𝐸 ᠶᠢ ᠳᠠᠭ ᠠᠷᠢᠭ ᠤᠯᠤᠨ 𝐸𝐻 ⊥ 𝑚 ᠢᠶᠠᠷ ᠪᠠᠢᠭ ᠤᠯᠪᠠ ᠯ᠂ ᠡᠭᠴᠡ ᠶᠢᠨ ᠰᠠᠭ ᠤᠷᠢ ᠨᠢ 𝐻 ᠂ 𝐴𝐶 ᠯᠤᠭ ᠡ ᠴᠡᠭ 𝐺 ᠳ᠋ᠦ ᠣᠭᠲᠣᠯ ᠣᠯᠴᠠᠵ ᠤ᠂ 𝐴𝐸 ,𝐸𝐶 ,𝐶𝐻 ,𝐴𝐻 ᠶᠢ ᠬᠣ ᠯᠪᠣᠨ ᠠ᠃ (1) ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠤᠨ ᠠᠨᠠ ᠯᠢᠰ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠨᠢ ( ) (2) ᠳᠥᠷᠪ ᠡᠯᠵᠢᠨ 𝐴𝐻𝐶𝐸 ᠶᠢᠨ ᠲᠠ ᠯᠠᠪᠠᠢ ᠨᠢ ᠬᠠᠮ ᠤᠭ ᠶᠡᠬᠡ ᠦᠶ ᠡᠰ᠂ ᠴ ᠡᠭ 𝐸 ᠦᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠣᠯ᠃ (3) ᠳᠡᠭᠡᠷ ᠡᠬᠢ (2) ᠤ ᠨ ᠨᠥᠬᠥ ᠴᠡᠯ ᠳᠣᠣᠷ ᠠ 𝐸𝐹 ᠶᠢ ᠬᠣᠯᠪᠣᠨ ᠠ᠂ ᠴᠡ ᠭ 𝑃 ᠨᠢ 𝑥 ᠲᠡ ᠩᠭᠡᠯ ᠢᠭ ᠳᠡᠭᠡᠷ ᠡᠬᠢ ᠨᠢᠭᠡ ᠬᠥᠳᠡ ᠯᠬᠦ ᠴᠡ ᠭ᠂ ᠬᠠᠶᠠᠯ ᠲᠠ ᠶᠢᠨ ᠱᠤᠭᠤ ᠮ ᠳᠡ ᠭᠡᠷ ᠡ 𝐹 ,𝐸 ,𝐵 ,𝑄 ᠢᠶᠡ ᠷ ᠣ ᠷᠣᠢ ᠶᠢᠨ ᠴᠡᠭ ᠂ 𝐸𝐹 ᠢᠶᠡᠷ ᠨᠢ ᠭᠡ ᠲᠠᠯ ᠠ ᠪᠣᠯ ᠭᠠᠭᠰᠠᠨ ᠳᠥᠷ ᠪᠡᠯᠵ ᠢᠨ ᠢ ᠵᠢᠰᠢᠬᠦᠯ ᠵᠢᠨ ᠪᠣᠯ ᠭᠠᠵᠤ ᠲᠠ ᠢᠯᠬᠦ 𝑄 ᠴᠡᠭ ᠣ ᠷᠣᠰᠢᠨ ᠠ ᠤᠤ︖ ᠬᠡᠷ ᠪᠡ ᠣᠷᠣ ᠰᠢᠪᠠᠯ ᠴᠡᠭ 𝑄 ᠶᠢᠨ ᠪᠠᠢᠷ ᠢᠰᠢᠯ ᠢ ᠰᠢᠭ ᠤᠳ ᠪᠢᠴᠢᠨ ᠭᠠᠷ ᠭ ᠠ᠂ ᠬᠡᠷ ᠪᠡ ᠣᠷᠣ ᠰᠢᠬ ᠤ ᠦᠭᠡ ᠢ ᠪᠣᠯ ᠰᠢᠯᠲᠠᠭ ᠠᠨ ᠢ ᠲᠣᠳ ᠣᠷᠬᠠᠢᠯ ᠠ᠃ 26. ᠲᠣᠭ ᠠᠨ ᠤ ᠤᠬ ᠠᠭᠠᠨ ᠤ ᠬᠢ ᠴᠢᠶᠡᠯ ᠳ᠋ᠦ ᠡᠢᠮᠤ ᠨᠢᠭᠡᠨ ᠰᠥᠪᠡ ᠭᠴᠢᠯ ᠡᠬᠦ ᠰ ᠡᠲᠤᠪ ᠪᠣᠢ᠄ ᠵᠢᠷᠤᠭ ᠮᠡᠲᠤ᠂ ᠦᠭ ᠭᠦᠭᠳᠡᠭᠰᠡᠨ ᠨᠢ ∆𝐴𝐵𝐶 ᠳ᠋ᠦ᠂ 𝐴𝐵 = 𝐴𝐶 = 𝑚 , 𝐵𝐶 = 𝑛 , ∠𝐵𝐴𝐶 = 𝑎( 0 𝑎 180° )᠂ ᠴᠡᠭ 𝑃 ᠨᠢ ᠬᠠᠪᠲᠠᠭ ᠠᠢ ᠶᠢᠨ ᠲᠣᠲ ᠣᠷ ᠠ 𝐴 ,𝐶 ᠯᠤᠭᠡ ᠲᠠᠪᠬᠤᠴ ᠠᠬᠤ ᠦᠭᠡᠢ ᠳᠤᠷ ᠠ ᠶᠢᠨ ᠨᠢᠭᠡ ᠴᠡᠭ ᠂ 𝐶𝑃 ᠶᠢ ᠬᠣᠯ ᠪᠣᠵᠤ ᠬᠡᠷᠴᠢ ᠮ 𝐶𝑃 ᠶᠢ ᠴᠡᠭ 𝑃 ᠶᠢ ᠲᠣᠭ ᠣᠷᠢᠭ ᠤᠯᠤᠨ ᠴᠠᠭ ᠤ ᠨ ᠵᠡᠭᠦᠦ ᠶᠢ ᠨ ᠳᠠᠭ ᠠᠤ 𝑎 ᠡᠷᠬᠢᠯ ᠳᠥᠬᠥ ᠯᠪᠡᠯ ᠬᠡᠷᠴᠢ ᠮ 𝑃𝐷 ᠶᠢ ᠣᠯᠣᠨ ᠠ᠃ 𝐶𝐷 ,𝐴𝑃 ᠶᠢ ᠬᠣᠯ ᠪᠣᠪᠠ ᠯ᠂ ᠴᠡ ᠭ 𝐸 ,𝐹 ᠨᠢ ᠲᠤᠰ ᠪᠦᠷᠢ 𝐵𝐶 ,𝐶𝐷 ᠶᠢᠨ ᠳᠤᠮᠳᠠ ᠴᠡᠭ ᠂ ᠰᠢᠯᠤᠭᠤᠨ 𝐴𝑃 ᠪᠣᠯ ᠣᠨ ᠰᠢᠯᠤᠭᠤᠨ 𝐸𝐹 ᠣᠭᠲ ᠣᠯᠣᠯ ᠴᠠᠭᠠ ᠳ ᠡᠭᠦᠰᠦ ᠭᠰᠡᠨ ᠨᠡ ᠯᠢᠶᠡᠳ ᠪᠠᠭ ᠠ ᠥᠨᠴᠥ ᠭ ᠢ 𝛽 ᠭᠡᠵᠤ ᠵᠢᠱᠢᠨ ᠡ᠃ 𝐸𝐹𝐴𝑃 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠪᠠ 𝛽 ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠪᠣᠯ ᠣᠨ 𝑚, 𝑛, 𝑎 ᠶᠢᠨ ᠬᠣᠯ ᠪᠣᠭᠳᠠᠯ ᠢ ᠬᠠᠢᠨ ᠰᠥᠪᠡ ᠭᠴᠢᠯ ᠡ᠃ ᠲᠠ ᠰᠤᠷᠤᠯ ᠴᠠᠯᠭ ᠠ ᠶᠢᠨ ᠪᠠᠭ ᠠ ᠳᠤᠭ ᠤᠢᠯ ᠠᠩ ᠤᠨ ᠬᠠᠢᠨ ᠰᠥ ᠪᠡᠭᠴᠢ ᠯᠡᠬᠦ ᠶᠠᠪᠤᠴ ᠠ ᠳ᠋ᠤ ᠣ ᠷᠣᠯᠴᠠ ᠵᠤ᠂ ᠪᠠᠰᠠ ᠳᠠᠷᠠᠭ ᠠᠬᠢ ᠡᠭᠦᠷ ᠭᠡ ᠶᠢ ᠪᠡᠶᠡᠯᠡᠬᠦ ᠯ᠃ (1) ᠬᠣᠭᠣᠰᠤ ᠨ ᠳ᠋ ᠤᠲᠠᠭ ᠯᠠ᠄ [ᠠᠰᠠᠭᠤᠳᠠ ᠯ ᠢ ᠢᠯᠡᠷ ᠡᠭᠦᠯᠬᠦ ] ᠰᠤᠷᠤᠭᠴᠢ ᠲᠤᠩᠭ ᠠᠯᠠᠭ α = 60° ᠤᠨ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠰᠤᠳᠤᠯ ᠪᠠ᠂ ᠵᠢᠷ ᠤᠭ 1 ᠮᠡᠲᠤ ᠂ 𝐸𝐹𝑃𝐴 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ β ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤ ᠨ ᠲᠣᠭ ᠠ ᠨᠢ ᠲᠤ ᠰᠪᠤᠷᠢ 𝐸𝐹𝑃𝐴 =_ __ , β =_ __ ᠰᠤᠷᠤᠭᠴᠢ ᠤᠯᠠᠭ ᠠᠨ α = 90° ᠤᠨ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠰᠤᠳᠤᠯ ᠪᠠ᠂ ᠵᠢᠷ ᠤᠭ 2 ᠮᠡᠲᠤ 𝐸𝐹𝑃𝐴 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ β ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤ ᠨ ᠲ ᠣᠭ ᠠ ᠨᠢ ᠲᠤ ᠰᠪᠤᠷᠢ 𝐸𝐹𝑃𝐴 =_ __ , β =_ __ [ᠬᠠᠷ ᠢᠴᠠᠭ ᠤᠯᠤᠨ ᠰᠥᠪᠡ ᠭᠴᠢᠯ ᠡᠬᠦ] ᠲᠡᠳᠡ ᠬᠠᠮᠳᠤ ᠪᠠᠷ α = 120° ᠤᠨ ᠪᠠᠢᠳᠠ ᠯ ᠢ ᠰᠤᠳᠤᠯ ᠪᠠ᠂ ᠵᠢᠷ ᠤᠭ 3 ᠮᠡ ᠲᠤ᠂ ᠪᠠᠰᠠ 𝐸𝐹𝑃𝐴 ᠶᠢᠨ ᠬᠡᠮᠵᠢᠭ ᠳᠡᠯ ᠪᠠ β ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤ ᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠣᠯᠪᠠ᠃ [ᠬᠤᠷᠢᠶᠠᠩᠭᠤᠢᠯ ᠠᠨ ᠳ᠋ ᠦᠩᠨᠡ ᠬᠦ] ᠬᠠᠮᠤᠭ ᠤᠨ ᠡᠴᠤ ᠰ ᠲᠤ ᠲᠡ ᠳᠡ ᠬᠠᠮᠳᠤ ᠪᠠᠷ 𝐸𝐹𝑃𝐴 = __ _ (𝑚 , 𝑛 ᠠᠭᠤᠯᠤᠭ ᠰᠠᠨ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢᠶᠡᠷ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡ ᠠᠭᠤ ᠯᠤᠭᠰᠠᠨ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡᠯ ᠢᠶᠡᠷ ᠢᠯᠡᠷ ᠭᠡᠢᠯ ᠡ )᠃ β= __ __ ( α ᠠᠭ ᠤᠯᠤᠭ ᠰᠠᠨ ᠢᠯᠡᠷ ᠬᠡᠢᠯ ᠡᠯ ᠢᠶᠡ ᠷ ᠢ ᠯᠡᠷᠭᠡᠢ ᠯᠡ ) ᠶ ᠢ ᠰᠤᠳᠤᠯᠤᠨ ᠣᠯᠪᠠ᠃ (2) α= 120° ᠦ ᠶᠡᠰ ᠦᠨ 𝐸𝐹𝑃𝐴 ᠶᠢᠨ ᠬᠡᠮᠵᠢ ᠭᠳᠡᠯ ᠪᠠ β ᠶᠢᠨ ᠭᠷᠠᠳᠦᠰ ᠤᠨ ᠲᠣᠭ ᠠ ᠶᠢ ᠣᠯ᠃